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Theorem flimclslem 23488
Description: Lemma for flimcls 23489. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
flimcls.2 𝐹 = (𝑋filGen(fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))
Assertion
Ref Expression
flimclslem ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑆 ∈ 𝐹 ∧ 𝐴 ∈ (𝐽 fLim 𝐹)))
Proof of Theorem flimclslem
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flimcls.2 . . 3 𝐹 = (𝑋filGen(fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))
2 topontop 22415 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
323ad2ant1 1134 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐽 ∈ Top)
4 eqid 2733 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ 𝐽
54neisspw 22611 . . . . . . . 8 (𝐽 ∈ Top β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝒫 βˆͺ 𝐽)
63, 5syl 17 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝒫 βˆͺ 𝐽)
7 toponuni 22416 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
873ad2ant1 1134 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑋 = βˆͺ 𝐽)
98pweqd 4620 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝒫 𝑋 = 𝒫 βˆͺ 𝐽)
106, 9sseqtrrd 4024 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝒫 𝑋)
11 toponmax 22428 . . . . . . . . . 10 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
12 elpw2g 5345 . . . . . . . . . 10 (𝑋 ∈ 𝐽 β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
1311, 12syl 17 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
1413biimpar 479 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 ∈ 𝒫 𝑋)
15143adant3 1133 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 ∈ 𝒫 𝑋)
1615snssd 4813 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ {𝑆} βŠ† 𝒫 𝑋)
1710, 16unssd 4187 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) βŠ† 𝒫 𝑋)
18 ssun2 4174 . . . . . 6 {𝑆} βŠ† (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})
19113ad2ant1 1134 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑋 ∈ 𝐽)
20 simp2 1138 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑋)
2119, 20ssexd 5325 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 ∈ V)
2221snn0d 4780 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ {𝑆} β‰  βˆ…)
23 ssn0 4401 . . . . . 6 (({𝑆} βŠ† (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) ∧ {𝑆} β‰  βˆ…) β†’ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) β‰  βˆ…)
2418, 22, 23sylancr 588 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) β‰  βˆ…)
2520, 8sseqtrd 4023 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† βˆͺ 𝐽)
26 simp3 1139 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†))
274neindisj 22621 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) ∧ (𝐴 ∈ ((clsβ€˜π½)β€˜π‘†) ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴}))) β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)
2827expr 458 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴}) β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…))
293, 25, 26, 28syl21anc 837 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴}) β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…))
3029imp 408 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)
31 elsni 4646 . . . . . . . . . . 11 (𝑦 ∈ {𝑆} β†’ 𝑦 = 𝑆)
3231ineq2d 4213 . . . . . . . . . 10 (𝑦 ∈ {𝑆} β†’ (π‘₯ ∩ 𝑦) = (π‘₯ ∩ 𝑆))
3332neeq1d 3001 . . . . . . . . 9 (𝑦 ∈ {𝑆} β†’ ((π‘₯ ∩ 𝑦) β‰  βˆ… ↔ (π‘₯ ∩ 𝑆) β‰  βˆ…))
3430, 33syl5ibrcom 246 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (𝑦 ∈ {𝑆} β†’ (π‘₯ ∩ 𝑦) β‰  βˆ…))
3534ralrimiv 3146 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ βˆ€π‘¦ ∈ {𝑆} (π‘₯ ∩ 𝑦) β‰  βˆ…)
3635ralrimiva 3147 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ βˆ€π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴})βˆ€π‘¦ ∈ {𝑆} (π‘₯ ∩ 𝑦) β‰  βˆ…)
37 simp1 1137 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
384clsss3 22563 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
393, 25, 38syl2anc 585 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
4039, 26sseldd 3984 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐴 ∈ βˆͺ 𝐽)
4140, 8eleqtrrd 2837 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐴 ∈ 𝑋)
4241snssd 4813 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ {𝐴} βŠ† 𝑋)
43 snnzg 4779 . . . . . . . . . 10 (𝐴 ∈ ((clsβ€˜π½)β€˜π‘†) β†’ {𝐴} β‰  βˆ…)
44433ad2ant3 1136 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ {𝐴} β‰  βˆ…)
45 neifil 23384 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ {𝐴} βŠ† 𝑋 ∧ {𝐴} β‰  βˆ…) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
4637, 42, 44, 45syl3anc 1372 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
47 filfbas 23352 . . . . . . . 8 (((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (fBasβ€˜π‘‹))
4846, 47syl 17 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (fBasβ€˜π‘‹))
49 ne0i 4335 . . . . . . . . . . 11 (𝐴 ∈ ((clsβ€˜π½)β€˜π‘†) β†’ ((clsβ€˜π½)β€˜π‘†) β‰  βˆ…)
50493ad2ant3 1136 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((clsβ€˜π½)β€˜π‘†) β‰  βˆ…)
51 cls0 22584 . . . . . . . . . . 11 (𝐽 ∈ Top β†’ ((clsβ€˜π½)β€˜βˆ…) = βˆ…)
523, 51syl 17 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((clsβ€˜π½)β€˜βˆ…) = βˆ…)
5350, 52neeqtrrd 3016 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((clsβ€˜π½)β€˜π‘†) β‰  ((clsβ€˜π½)β€˜βˆ…))
54 fveq2 6892 . . . . . . . . . 10 (𝑆 = βˆ… β†’ ((clsβ€˜π½)β€˜π‘†) = ((clsβ€˜π½)β€˜βˆ…))
5554necon3i 2974 . . . . . . . . 9 (((clsβ€˜π½)β€˜π‘†) β‰  ((clsβ€˜π½)β€˜βˆ…) β†’ 𝑆 β‰  βˆ…)
5653, 55syl 17 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 β‰  βˆ…)
57 snfbas 23370 . . . . . . . 8 ((𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ… ∧ 𝑋 ∈ 𝐽) β†’ {𝑆} ∈ (fBasβ€˜π‘‹))
5820, 56, 19, 57syl3anc 1372 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ {𝑆} ∈ (fBasβ€˜π‘‹))
59 fbunfip 23373 . . . . . . 7 ((((neiβ€˜π½)β€˜{𝐴}) ∈ (fBasβ€˜π‘‹) ∧ {𝑆} ∈ (fBasβ€˜π‘‹)) β†’ (Β¬ βˆ… ∈ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ↔ βˆ€π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴})βˆ€π‘¦ ∈ {𝑆} (π‘₯ ∩ 𝑦) β‰  βˆ…))
6048, 58, 59syl2anc 585 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (Β¬ βˆ… ∈ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ↔ βˆ€π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴})βˆ€π‘¦ ∈ {𝑆} (π‘₯ ∩ 𝑦) β‰  βˆ…))
6136, 60mpbird 257 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ Β¬ βˆ… ∈ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))
62 fsubbas 23371 . . . . . 6 (𝑋 ∈ 𝐽 β†’ ((fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ∈ (fBasβ€˜π‘‹) ↔ ((((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) βŠ† 𝒫 𝑋 ∧ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) β‰  βˆ… ∧ Β¬ βˆ… ∈ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))))
6319, 62syl 17 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ∈ (fBasβ€˜π‘‹) ↔ ((((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) βŠ† 𝒫 𝑋 ∧ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) β‰  βˆ… ∧ Β¬ βˆ… ∈ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))))
6417, 24, 61, 63mpbir3and 1343 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ∈ (fBasβ€˜π‘‹))
65 fgcl 23382 . . . 4 ((fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ∈ (fBasβ€˜π‘‹) β†’ (𝑋filGen(fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}))) ∈ (Filβ€˜π‘‹))
6664, 65syl 17 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (𝑋filGen(fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}))) ∈ (Filβ€˜π‘‹))
671, 66eqeltrid 2838 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
68 fvex 6905 . . . . . 6 ((neiβ€˜π½)β€˜{𝐴}) ∈ V
69 snex 5432 . . . . . 6 {𝑆} ∈ V
7068, 69unex 7733 . . . . 5 (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) ∈ V
71 ssfii 9414 . . . . 5 ((((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) ∈ V β†’ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) βŠ† (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))
7270, 71ax-mp 5 . . . 4 (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) βŠ† (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}))
73 ssfg 23376 . . . . . 6 ((fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ∈ (fBasβ€˜π‘‹) β†’ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) βŠ† (𝑋filGen(fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}))))
7464, 73syl 17 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) βŠ† (𝑋filGen(fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}))))
7574, 1sseqtrrdi 4034 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) βŠ† 𝐹)
7672, 75sstrid 3994 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) βŠ† 𝐹)
77 snssg 4788 . . . . 5 (𝑆 ∈ V β†’ (𝑆 ∈ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) ↔ {𝑆} βŠ† (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))
7821, 77syl 17 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (𝑆 ∈ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) ↔ {𝑆} βŠ† (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))
7918, 78mpbiri 258 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 ∈ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}))
8076, 79sseldd 3984 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 ∈ 𝐹)
8176unssad 4188 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)
82 elflim 23475 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
8337, 67, 82syl2anc 585 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
8441, 81, 83mpbir2and 712 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐴 ∈ (𝐽 fLim 𝐹))
8567, 80, 843jca 1129 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑆 ∈ 𝐹 ∧ 𝐴 ∈ (𝐽 fLim 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  Vcvv 3475   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7409  ficfi 9405  fBascfbas 20932  filGencfg 20933  Topctop 22395  TopOnctopon 22412  clsccl 22522  neicnei 22601  Filcfil 23349   fLim cflim 23438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1o 8466  df-er 8703  df-en 8940  df-fin 8943  df-fi 9406  df-fbas 20941  df-fg 20942  df-top 22396  df-topon 22413  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-fil 23350  df-flim 23443
This theorem is referenced by:  flimcls  23489
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